Assertion (A): A wave represented by y ( x , t ) = A sin ( k x − ω t + ϕ ) is a progressive wave moving in the positive x-direction. Reason (R): For a progressive wave, the phase ( k x − ω t + ϕ ) rem

Question

Assertion (A): A wave represented by y ( x , t ) = A sin ( k x − ω t + ϕ ) is a progressive wave moving in the positive x-direction. Reason (R): For a progressive wave, the phase ( k x − ω t + ϕ ) remains constant for a point moving with the wave velocity. A. Both A and R are true and R is the correct explanation of A. B. Both A and R are true but R is not the correct explanation of A. C. A is true but R is false. D. A is false but R is true.

TestHub · Accepted Answer

For a progressive wave, the phase of the wave remains constant for an observer moving with the wave velocity. If the phase $Φ = k x - ω t + ϕ$ is constant, then $d Φ/ d t = 0$ .

This implies $k \frac{d x}{d t} - ω = 0$ , so the wave velocity $v_{p} = \frac{d x}{d t} = \frac{ω}{k}$ . Since $ω$ and $k$ are positive, the wave moves in the positive x-direction. Both A and R are true, and R provides the fundamental definition that explains the direction of wave propagation.

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